Frequency of $a$-points for the fifth and the third Painlevé transcendents in a sector

Abstract

For the fifth Painlevé transcendents in a sector, under the condition that the values taken along some curve tending to infinity are bounded away from 1 and another specified complex number, we present new upper estimates for the number of $a$-points including poles and for the growth order. As far as we are concerned with the known asymptotic solutions of the fifth Painlevé equation, this condition is easily checked, and our results are applicable to almost all of them. About concrete examples we discuss the frequency of $a$-points, the equi-distribution property and the growth order. Our method works on the third Painlevé transcendents as well, yielding an analogous result.